Passification-Based Adaptive Control of Linear

Systems: Robustness Issues

Dimitri Peaucelle†, Alexander L. Fradkov‡ and Boris Andrievsky‡

† LAAS-CNRS, Universite de Toulouse7, av. du colonel Roche, 31077 Toulouse, FRANCE

Email: peaucelle@laas.fr‡ Institute for Problems of Mechanical Engineering of RAS

61 Bolshoy av. V.O., St Petersburg, 199178, RUSSIA

July 29, 2008

Abstract

Passivity is a widely used concept in control theory having lead tomany significant results. The manuscript concentrates on one charac-teristic of passivity, namely passification-based adaptive control. Thisconcept applies to MIMO systems for which exists a combination of out-puts that renders the open-loop system hyper-minimum-phase. Undersuch assumptions, the system may be passified by both high-gain staticoutput-feedback and by a particular adaptive control algorithm. This lastcontrol law is modified here to guarantee its coefficients to be bounded.The contribution of the paper is to investigate its robustness with respectto parametric uncertainty. Time response characteristics are illustratedon examples including realistic situations with noisy output and saturatedinput. Theoretical results are formulated as linear matrix inequalities andcan hence be readily solved with semi-definite programming solvers.Keywords: Passivity, Robustness, Adaptive Control, LMI

1 INTRODUCTION

One of a variety of adaptive control approaches is the so-called passification basedadaptive control (PBAC). First introduced in 1974 for linear systems [9, 10, 1]and later extended to nonlinear systems [29, 30, 18, 13] it provides efficient de-sign procedures and simple controller structures. Compared to adaptive schemeswith combined parameter estimation and controller tuning [16, 4], PBAC needsno estimation and tuning is performed via a simple differential equation. Forthis reason it is also called simplified adaptive control in [19].

In PBAC framework, passification (sometimes called passivation) is under-stood as finding a state or output feedback control rendering the closed loopsystem passive [29, 20]. In the paper two passification control strategies are

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considered. More precisely, both strategies are required to be G-passifing [11].That is, closed-loop passivity should be attested not for the entire measurementoutput vector but for a linear combination defined by a matrix G. As G maynot be square, the results apply for non-square systems. An algorithm based onBilinear Matrix Inequalities (BMIs) is provided for the design of G matrices.

Among applications areas of PBAC are process control [2], flight control [3,12] and irrigation systems [32]. But many more applications would be possibleif providing methods and proofs for assessing robustness of PBAC with respectto disturbances and parametric uncertainties. These long standing issues (seee.g. [8, 27]) are the main questions addressed in the paper.

Lack of robustness with respect to disturbances is noticed for many adaptivealgorithms and is often characterized by diverging adaptation parameters. Manypractical solutions have been proposed and one of the most popular ones isintroducing a negative feedback into the adaptation differential equation.It wasfirst employed in 1971 [24, 7] and rediscovered later [8, 15]. In the presentpaper, such negative feedback on the controller parameters is proposed basedon a dead-zone type function. It is shown to have good behavior to disturbancesand moreover it guarantees convergence to bounded equilibrium points. It istherefore called a Bounded Passification-Based Adaptive Control (BPBAC).

But the central result of the paper is to address robustness with respectto parametric uncertainty. While robustness is often complex to evaluate ina non-linear context, many results exist for linear systems [6, 17, 28]. Theproposed approach therefore relies on a theoretical result relating linear staticoutput-feedback (SOF) and BPBAC. Namely it is demonstrated that BPBAChas robust closed-loop passivity properties if one can prove the existence of apassifying parameter-dependent SOF with bounded gains. Limiting the studyto state-space modeled linear time-invariant systems with uncertainties on thethe matrix of dynamics A, a method is provided for such parameter-dependentSOF design. The result is derived from nowadays widely known Linear MatrixInequality (LMI) techniques [6] which are adapted for the passification problemfollowing the first results of [25, 26].

The outline of the manuscript is as follows. In the next section the robust G-passification problem is stated and the class of systems for which it is solved arepresented. Section 3 is then devoted to LMI-based results for robust parameter-dependent SOF design. The obtained numerically testable conditions are provedin section 4 to attest robustness of the BPBAC. Section 5 gives the BMI algo-rithm for selecting appropriate G matrices. In section 6 all exposed results areapplied to a numerical example and realistic closed-loop simulations are pro-vided. All numerical calculations are performed in the MATLAB environment.YALMIP [23] is used to enter LMIs and BMIs. Semi-definite programming prob-lems are solved with SeDuMi [31] and BMI problems are solved with PenBMI[21, 22].

Notations: Rm×n and Cm×n are the sets ofm-by-n real and complex matricesrespectively. For a matrix A, the notations A(i,j) indicates the element of rowi and column j. AT is the transpose of the matrix A and A∗ is its transposeconjugate. Tr(A) is the trace of matrix A. 1 and 0 are respectively the identity

2

and the zero matrices of appropriate dimensions depending on the context. ForHermitian matrices, A > (≥)B if and only if A−B is positive (semi) definite.

2 PROBLEM FORMULATION

Control of uncertain LTI systems is considered. The systems are represented instate-space by {

x(t) = A(ξ)x(t) +Bu(t)y(t) = Cx(t) (1)

where x(t) ∈ Cn is the state, u(t) ∈ Cm is the vector of control inputs andy(t) ∈ Cl is the vector of measured outputs. ξ is a constant parameter describingdependency with respect to operating conditions. This parameter is decomposedas a sum of two terms ξ = θ+∆ where θ stands for an estimate of the operationpoint while ∆ is an uncertainty on the conditions. The matrix A(θ + ∆) isassumed rationally dependent with respect to ∆, which dependency can begenerically written as:

A(θ + ∆) = A0(θ) +B∆(θ)∆(1−D∆(θ)∆)−1C∆(θ) .

The uncertainty matrix ∆ is assumed constant full-block norm-bounded, ∆ ∈∆ρ(θ) where

∆ρ = {∆ ∈ Cm∆×l∆ : ∆∗∆ ≤ ρ21} (2)

and ρ(θ) is the radius of the ball of uncertainties. It depends of the operatingpoint. The larger is ρ, the bigger is the domain of admissible uncertainties.Many other types of uncertainties may be considered in the same framework asexposed in the following. Norm-bounded type is adopted to simplify presenta-tion of results.

The B and C matrices are assumed to be exactly known and full rank. Thisassumption restricts the paper application to systems with uncertainties onlyon the system dynamics. Difficulty to extend the exposed results for systemswith uncertainties on B and C matrices are discussed further in the paper. Suchextensions are planned for future research.

The defined uncertain model is said to be a Linear Fractional Transform(LFT), build as the feedback connection of the uncertain matrix (w∆ = ∆z∆)with the linear system x = A0(θ)x+B∆(θ)w∆ +Bu

z∆ = C∆(θ)x+D∆(θ)w∆

y = Cx(3)

The LFT is assumed to be well-posed, that is (1−D∆(θ)∆) is non-singular forall admissible uncertainties ∆ ∈ ∆ρ(θ) and A(θ,∆) lies in a bounded set.

Two control strategies are adopted and compared. One is parameter-dependentstatic output-feedback (SOF)

u(t) = F (θ)y(t) + v(t) (4)

3

which supposes to have measurements of the operating point parameters (or tohave access to a good estimate); the second, is adaptive control defined as

u(t) = K(t)y(t) + v(t)K(t) = −Gy(t)y∗(t)Γ− φ(K(t))Γ

(5)

where φ is any dead-zone type function satisfying property P(B).

Definition 1 Let B ⊂ Cm×l be a compact set of complex valued matrices. Thefunction φ(K) : Cm×l → Cm×l, it is said to satisfy property P(B) if

• the function is zero valued inside B:

φ(K) = 0 , ∀K ∈ B

• and, outside B, makes strictly positive the following scalar product:

Tr (φ(K)(K − F )∗) > 0 , ∀K ∈ Cm×l\B , ∀F ∈ B .

A sub-case of adaptive control (5) is when φ(K) is identically zero for allK ∈ Cm×l. It corresponds to an infinite dead-zone where B = Cm×l. Thisinfinite dead-zone case is exactly the control strategy adopted in [9, 19] andis known to diverge in the presence of disturbances on the measurements y(t).A classical practical solution to prevent the gain K(t) for diverging is to adda term such as −φ(K(t)) (see for example [8, 15, 4]) which will force K(t) tokeep close to a region B of implementable gains. The idea of using dead-zonefunctions rather than other proportional gain regularizations is not new (see forexample [14] for adaptive observer case). The present contribution is to considerdead-zone regularization in a control framework and to prove that propertiesholding for φ identically zero are kept true when the dead-zone type function isintroduced.

The provided results prove robust closed-loop stability and passivity of (1)with adaptive control (5) for multiple operating points θ ∈ {θ1 . . . θN} whilemaximizing the size ρ(θi) of admissible uncertainties ∆ around each one ofthese values. Robustness of the adaptive control is then proved for ξ lying inthe union of all ”balls” centered at the operating points θi thus giving an innerapproximation of the actual region of admissible parameters ξ ∈ Ξ (see figure 1).Results are provided assuming the systems matrices of equations (3) are givenand the region B is set a priori by practical implementation considerations.

Before getting into details, the definition of passivity on which are based allresults is stated. It is done in a non-linear context because it corresponds to thesituation when (1) is in closed-loop with the adaptive non-linear control (5).

Consider a closed-loop non-linear uncertain system{η = f(η,∆) + g(η)vy = h(η) (6)

4

Ξ

θ

ρ(θ)

Figure 1: Inner approximation of the admissible domains Ξ by finite number of”balls”

where η =(x∗ K∗ )∗ is the state, v is the input, y is the measurements, ∆ is

an uncertainty, f , g and h are smooth functions. Let G be a prespecified m× l-matrix. Extending the definitions of [11] to uncertain systems and insistingof the sub-part x of the state that corresponds to the original system to becontrolled, the following definition is stated.

Definition 2 The system (6) is called robustly globally x-strictly G-passive iffor any ∆ ∈ ∆ there exists a nonnegative scalar function V (η,∆) (storagefunction) and a scalar function γ(x,∆) (strictly positive for all x 6= 0) such that

V (η(t),∆) ≤ V (η(0),∆) +t∫0

[v(τ)∗Gy(τ)− γ(x(τ),∆)] dτ (7)

holds for all t ≥ 0 and all solutions of the system (6).

The adjective ”robustly” indicates passivity should hold for all admissiblevalues of the uncertainty ∆ ∈ ∆, when omitted it indicate the property holdsfor only one considered value of ∆. The adjective ”globally” indicates passivitydoes not depend on the initial conditions. As usual when dealing with controlof linear systems, specifications are always global and the adjective is removedto alleviate. The adjective ”strict” indicates that not only the system is passive,but for zero inputs the sub-state x(t) converges to zero. As usually for adaptivecontrol no convergence is required for the sub-state K(t). On the contrary,except if the open-loop system is already asymptotically stable and passive,K(t) has to be different from zero for stability to hold. ”G-passivity” indicatesthat the usual passivity properties hold with respect to a linear combination ofthe closed-loop outputs characterized by the matrix G. This notion allows toextend results to many more systems than when taking G = 1, in particular

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systems may not be square. Note that design of G-passifying controllers K for asystem Σ (i.e. find K such that the closed-loop with linearly combined outputsG(Σ ? K) is passive) is not equivalent to passification of the ”squared” systemGΣ (i.e. find K such that the closed-loop (GΣ) ?K is passive). G-passificationas the advantage to be more general and it preserves the number of adjustableparameters potentially giving better transient responses, see [1].

3 ROBUST PASSIFYING SOF DESIGN

Static-output feedback design is considered at first with an additional constrainton modulus-bounded entries. Let F ∈ Rm×l be a real valued matrix with strictlypositives entries (F (i,j) > 0). The sub-set BF ⊂ Cm×l is defined as the set ofcomplex valued matrices with entries bounded in modulus by the entries of F :

F ∈ BF ⇔ |F(i,j)| ≤ F (i,j) .

Based on results of [10, 11], an LMI test for robust G-passifiability wasrecently published in [25]. Slightly modified to take into account uncertain setswith ρ 6= 1, and including bounds on the controller gain, it writes as follows.

Theorem 1 Let G ∈ Cm×l a given matrix, if there exists a solution H(θ) ∈Cn×n, F (θ) ∈ Cm×l to the LMI constraints

H(θ) > 0 , H(θ)B = C∗G∗ , (8)[H(θ)A0(θ) +A∗0(θ)H(θ) + C∗(G∗F (θ) + F ∗(θ)G)C H(θ)B∆(θ)

B∗∆(θ)H(θ) 0

]+

[C∗

∆(θ) 0D∗

∆(θ) 1

] [ρ2(θ)1 0

0 −1

] [C∆(θ) D∆(θ)

0 1

]< 0

(9)

and for (i, j) ∈ {1 . . .m} × {1 . . . l}[F

2

(i,j) F(i,j)(θ)F ∗

(i,j)(θ) 1

]≥ 0 , (10)

then F (θ) ∈ BF and the uncertain system (1) at operating point θ is robustlyx-strictly G-passified via SOF u(t) = F (θ)y(t) + v(t) for all uncertainties ∆ ∈∆ρ(θ).

Proof : The fact that F (θ) ∈ BF is immediate applying a Schur complementargument to the inequalities (10). (9) being a strict inequality, there exists apositive scalar ε such that

L(9) ≤[−ε1 00 0

]

6

where L(9) stands for the left-hand side of inequality (9). Pre and post multiplythe obtained inequality by the vector

(x∗ w∗∆

)and its transpose respectively

to get

2x∗H(θ)(A0(θ)x+B∆(θ)w∆) + 2x∗C∗G∗F (θ)y + εx∗x ≤ w∗∆w∆ − ρ2(θ)z∗∆z∆ .(11)

By definition of the uncertainty, the right-hand side of this inequality is negative.Due to (8) and feedback control law (4) one gets x∗C∗G∗F (θ)y = x∗H(θ)Bu−y∗G∗v. Hence (11) implies

2x∗H(θ)x ≤ 2v∗Gy − εx∗x .

Taking its integral over time gives (7) where V (x) = x∗H(θ)x and γ(x) =1/2εx∗x. �

Although the current paper considers only full-block norm-bounded uncer-tainties to limit the presentation complexity, note that the result of Theorem 1extends easily to many more types of uncertainties. Following methodology of

papers such as [17, 28] it simply needs to modify the[ρ2(θ)1 0

0 −1

]matrix

accordingly to the uncertainty description.Some characteristics of Theorem 1:

• The equality constraint of (8) is a strong limiting constraint that, exceptmaybe in very special cases, cannot hold for uncertainty dependent matri-ces B and C. The assumption that B and C are exactly known is at thisstage necessary. Future work will be devoted to removing this limitation.

• The constraints (8), (9) and (10) are linear with respect to the decisionvariables H(θ) and F (θ) hence finding such a solution can be done effi-ciently using semi-definite programming solvers such as those interfacedin YALMIP [23].

• The LMI conditions are also linear with respect to ρ(θ). Therefore, thesolvers can as well maximize ρ(θ) for increasing the uncertainty domainson which robustness is guaranteed.

Solving separately optimization problems for several operating points θ ∈θ = {θ1 . . . θN} gives a sub-set of operating points θ ⊂ θ for which LMIs ofTherorem 1 are feasible. Associated to this sub-set are sequences of boundedfeedback gains {F (θ) ∈ BF }θ∈θ

and associated bounds on uncertainties {ρ(θ)}θ∈θ.Define the overall set of obtained operating points:

ΞF ={

ξ = θ + ∆(θ) : θ ∈ θ , ∆(θ) ∈ ∆ρ(θ)

}.

Denote ΞF the set of all operating conditions ξ such that system (1) is G-passifiable via a parameter-dependent SOF (4) with gains bounded in BF . Theset ΞF obtained by solving a finite number of LMI conditions, is trivially aninner approximation of ΞF . Moreover, taking a sufficiently thin discretization

7

θ of the parameter space, the inner approximation can be made as exact aswanted. Without entering into details on building and optimizing an algorithmfor picking appropriately the elements of θ, note that such algorithm would needonly to solve at each tested point LMIs of relatively low dimensions with fewvariables. All the calculations being done off-line, the overall computationalburden is relatively limited.

4 ROBUST PASSIFYING ADAPTIVE CON-TROL

The previous section concludes with the possibility to compute a set ΞF suchthat for all operating conditions ξ ∈ ΞF there exists a SOF control gain F (ξ) ∈BF that G-passifes system (1), moreover the such gains are known and belongto the finite set {F (θ)}θ∈θ. But the underlying parameter-dependent controlstrategy is not recommended. In case ξ is indeed constant, implementationwould mean to be able to measure it (or to build a sufficiently fast estimatorto get an appropriate value before the system diverges) and couple that witha look-up table for F (ξ) ∈ {F (θ)}θ∈θ selection. In case ξ is not constant butslowly varying, the control strategy may totally fail because of switching betweencontrollers. Rather than entering such complex, computation and memory de-manding strategies, direct adaptive control is adopted in the following.

Before stating the main result of the paper, equilibrium points of the closed-loop with adaptive control are investigated. The closed-loop system is non-linearwith n+ml states corresponding to the n original states x(t) of (1) and the mlelements of the matrix K(t) of system (5). Define the set

E = { (xe,Ke) : xe = 0 , Ke = F ∈ B } .

As φ(F ) = 0 for all F ∈ B, assuming zero input v(t) = 0, the states included inE are such that

x = 0 , K = 0 .

E is a set of equilibrium points for system (1) with adaptive control (5). For agiven operating condition ξ, around the equilibrium point (xe = 0,Ke = F (ξ)),the linearized model is such that

x(t) = A(ξ)x(t) +BKeCx(t) +BK(t)Cxe = (A(ξ) +BF (ξ)C)x(t) .

First Lyapunov’s theorem indicates that the equilibrium point is locally unstableif A(ξ)+BF (ξ)C has a positive eigenvalue. This result indicates that E cannotbe an asymptotically stable set if B does not contain a stabilizing SOF gain.Moreover, if E is proved to be a globally asymptotically stable set and assumingthe initial conditions are not at an unstable equilibrium, the system states canonly converge, if it does converge to a fixed point (which is the case in practice),to a value K(+∞) = F (ξ) which defines a stabilizing SOF gain for system (1).

The central result is now formulated.

8

Theorem 2 Let G ∈ Cm×l a given matrix, the following condition (i) implies(ii):

(i) There exists a parameter-dependent m × l matrix F (ξ) ∈ B such thatsystem (1) with the SOF (4) is robustly x-strictly G-passive for all ξ ∈ Ξ.

(ii) For any Hermitian positive definite matrix Γ ∈ Cl×l and any function φsatisfying property P(B), the choice (5) is a time-varying output-feedbackthat renders system (1) robustly x-strictly G-passive for all ξ ∈ Ξ and, incase of zero input v(t) = 0, the set E is globally asymptotically stable.

Proof : Take any ξ ∈ Ξ, let F (ξ) ∈ B be an asymptotically stabilizinggain and assume it x-strictly G-passifies the system. According to [11], this isequivalent to the existence of a quadratic storage function V (x, ξ) = x∗H(ξ)xand a positive scalar γ(ξ) > 0 fulfilling the matrix inequalities

H(ξ) = H∗(ξ) > 0 : H(ξ)B = C∗G∗

H(ξ)A(ξ, F ) +A∗(ξ, F )H(ξ) < −2γ(ξ)1 (12)

where A(ξ, F ) = A(ξ) + BF (ξ)C is the closed-loop dynamics matrix. Let anyHermitian positive definite matrix Γ > 0 and let the output-feedback law (5).Consider the following storage function

V (x,K, ξ) =12x∗H(ξ)x+

12Tr

((K − F (ξ))Γ−1(K − F (ξ))∗

). (13)

Along the trajectories of (1) with the control law (5) the derivatives of V (x,K, ξ)write

V (x,K, ξ) = x∗H(ξ)(A(ξ)x+BKy +Bv) + Tr((K − F (ξ))Γ−1K∗

).

Add and subtract x∗H(ξ)BF (ξ)y in the equation to get

V (x,K, ξ) = x∗H(ξ)(A(ξ)x+BF (ξ)y +Bv) + x∗H(ξ)B(K − F (ξ))y+Tr

((K − F (ξ))Γ−1K∗

)which, taking y = Cx, H(ξ)B = C∗G∗ and K as in (5), reads

V (x,K, ξ) = x∗H(ξ)A(ξ, F )x+ y∗Gv + y∗G∗(K − F (ξ))y−Tr ((K − F (ξ))yy∗G∗)− Tr ((K − F (ξ))φ(K)∗) .

As Tr(M1M2) = Tr(M2M1) one gets that

y∗G∗(K − F (ξ))y − Tr ((K − F (ξ))yy∗G∗)= y∗G∗(K − F (ξ))y − Tr (y∗G∗(K − F (ξ))y)= 0

therefore, due to (12), the derivative of the Lyapunov function is over-boundedas

V (x,K, ξ) ≤ −γ(ξ)||x||2 + y∗Gv − Tr (φ(K)(K − F (ξ))∗) . (14)

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Since F (ξ) ∈ B and due to property P(B), one gets that for zero input v = 0 andfor any vector (x,K) not belonging to E the time derivative of the Lyapunovfunction V is strictly negative:

V (x,K, ξ) < 0 , v = 0 , ∀(x,K) /∈ E

which proves global asymptotically stability of E . Taking the integral over timeof (14), one gets

V (x(t),K(t), ξ) ≤ V (x(0),K(0), ξ) +

t∫0

[v(τ)∗Gy(τ)− γ(ξ)||x(τ)||2

]dτ

which proves x-strict G-passivity for the closed-loop. �Some characteristics of Theorem 2:

• The time-varying control (5) is called the Bounded Passification-BasedAdaptive Controller (BPBAC). The values of the gain K(t) are automat-ically tuned given the measures y(t), it adapts whatever the values of theuncertain parameters ξ and thanks to the dead-zone function φ, the con-troller gains are constrained to converge in a bounded set. Moreover, itis shown that the BPBAC converges asymptotically to a stabilizing SOFgain.

• The BPBAC is not claimed to ensure robustness for any parameter ξ(hardly believable in practice). But applying results of section 3, in caseB is of the type BF , admissible sets ΞF fulfilling property (i) of Theorem2 may be computed with LMI tools.

5 BMI DESIGN OF G MATRICES

All results exposed up to this point assume a given G ∈ Cm×l matrix for whichthe system is desired to be closed-loop G-passive. Trivially, not all matricesare suitable. For that purpose (except if physical considerations indicate agood choice or if techniques of [5] apply) there is a need for a guide to choosethe matrix G. Ideally, one may formulate the problem as finding G such thatthe set ΞF is maximal. Not only such problem would be difficult to statemathematically but as seen in the following it would be much complex froma numerical point of view. Therefore the paper adopts a sub-optimal strategyconsisting of designing G such that ΞF is non-empty and contains at least onevalue ξ0 to be chosen a priori.

In [11, Corollary 3] it is proved that if there exists a SOF gain that G-passifies the system (1) at operation point ξ0 then the choice F (ξ0) = −k(ξ0)G,for a sufficently large value of k(ξ0), is also a G-passifying gain. This result isquite common in passivity context and is known as high-gain control. For ourpurpose, it allows to simplify the search of G. Indeed G-passifiability conditions

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(12) write for F (ξ0) = −k(ξ0)G:

H(ξ0) = H∗(ξ0) > 0 : H(ξ0)B = C∗G∗

H(ξ0)A(ξ0) +A∗(ξ0)H(ξ0) < 2k(ξ0)C∗G∗GC(15)

This problem is non-convex and, if k(ξ0) is fixed a priori, it is a problem con-strained by Bilinear Matrix Inequalities (BMIs). It may nevertheless be solvedusing the PenBMI solver [21, 22]. Of course, since BMI problems are not con-vex, there is no guarantee for finding G even when it exists. However, as testedon several examples, PenBMI does succeed efficiently.

The design procedure we have adopted for G-design is

• Choose a large value of k(ξ0);

• Choose an upper bound onH(ξ0) (we took h = 1) for scaling the solutions;

• Declare in YALMIP the following BMI problem where H(ξ0), G and t arethe variables

h1 > H(ξ0) > 0 , H(ξ0)B = C∗G∗ , t > −1 (16)

H(ξ0)A+A∗H(ξ0)− k(ξ0)C∗G∗GC < t1 (17)[F

2

(i,j) −k(ξ0)G− k(ξ0)G∗ 1

]≥ 0 , (18)

• Minimize t using PenBMI, if it returns t < 0, the procedure succeeded.

Some characteristics of the procedure:

• The only bilinear term in the inequalities is G∗G. Every other term inlinear, which may explain the relatively good behavior of PenBMI for theseBMIs.

• The last inequality (18) is added to the optimization problem to guaranteethe existence of at least one gain F in the set BF .

• Once a matrix G has been found, any matrix proportional to that is alsosatisfactory for the considered problem. We recommend for numericalreasons to choose k(ξ0)G.

6 NUMERICAL EXAMPLE

Consider the linearized fourth-order model of lateral dynamics for an autonomousaircraft including model of actuator dynamics, presented in [12]. The nominalmodel is defined for a medium value of the flight altitude ξ0 = 5km. The mea-sured plant output y(t) is a vector of the yaw angle ψ(t), yaw angular rate r(t)and the rudder deflection angle δr(t): y(t) = [ψ, r, δr]∗. The control input of theplant is the rudder servo command signal, i.e. n = 4, m = 1, l = 3.

11

Parameters of the nominal state-space model (1) in this case are as follows:

A(ξ0) =

0 1 0 00 0 1 00 12 −0.6 5.00 0 0 −20

, B =

00020

, C =

1 2 0 00 1 2 00 0 0 1

and dependency with respect to fight altitude is given by

B∆ =[

0 0 0.2 0]T

, D∆ = 0C∆ =

[0 −7.5 0.7 −4.5

].

Note that for this data some coefficients of the matrix A(ξ) vary in an order ofmagnitude when ξ varies from 0 to 10.

6.1 Design of G matrices

This design procedure is applied with two choices of k:

k1 = 102 , k2 = 104

and with F =[

10 10 10], i.e. all coefficients of the control gain should be

norm bounded by 10. It gives the following two admissible values of G

G1PenBMI = 10−2[

10 6.14 3.55]

G2PenBMI = 10−4[

10 10 4.87].

The computation time of PenBMI is less than half a second for the example(Sunblade 150 computer). The scaled values

G1 =[

10 6.14 3.55], G2 =

[10 10 4.87

]are admissible as well. These are used next.

6.2 Computation of the admissible set of flight altitudes

LMIs of Theorem 1 are solved with F =[

10 10 10], for each matrix G1 and

G2 and for sequences of values θi. The latest version of SeDuMi [31] (SeDuMi1.1 available at http://sedumi.mcmaster.ca/) is used to solve the LMIs alongwith the parser YALMIP [23]. The computation time of each individual LMIproblem is less than half a second (Sunblade 150 computer). No numericalproblems are encountered.

The results are given in Tables 1 and 2. The obtained admissible uncertaintysets ΞF are the unions of the discs centered at θi with radius ρ(θi) plotted inFigure 2.

One can notice that depending on the flight conditions, the SOF gains arequite different. Although there may be a single SOF gain G-passifying thesystem for the entire sets ΞF , the results tend to indicate that such gain would

12

Table 1: Results of Theorem 1 for G1

θi ρ(θi) F (θi)0 1.811

[−9.5880 −6.2087 −10

]3 1.8557

[−9.8017 −9.3363 −10

]6 1.6791

[−10 −10 −7.7524

]8 1.2443

[−10 −10 −3.8156

]9.2 0.4667

[−6.4047 −6.2846 0.1371

]9.6 0.0753

[−1.0534 −1.1990 0.8271

]Table 2: Results of Theorem 1 for G2

θi ρ(θi) F (θi)0 2.0850

[−9.2301 −7.2487 −10

]4 2.1685

[−9.9957 −9.9653 −10

]7 1.7141

[−10 −10 −5.7759

]9 0.5217

[−4.7837 −6.0888 −0.0732

]9.5 0.0248

[−0.5089 −0.7243 0.7936

]

!8 !6 !4 !2 0 2 4 6!2

!1.5

!1

!0.5

0

0.5

1

1.5

2

!8 !6 !4 !2 0 2 4 6!2.5

!2

!1.5

!1

!0.5

0

0.5

1

1.5

2

2.5

Figure 2: Unions of admissible uncertain sets for G1 and G2 respectively

have coefficients with norm larger than 10. When limiting the control gain it istherefore needed to have an auto-tuning rule to adapt the gains depending onthe flight altitudes. BPBAC is applied for this purpose in the following.

Note as well that uncertainties are assumed complex in Theorem 1. This iswhy the regions are discs of the complex plane. For practical purpose onlythe real part of these sets are of interest. The result is that there existsa bounded parameter-dependent SOF such that the system is robustly G1-passsifiable for all real valued flight altitude ξ ∈ [0 9.6753] and G2-passifiablefor all ξ ∈ [0 9.5248]. Theorem 2 guarantees that for these sets the BPBAC (5)asymptotically stabilizes the system and makes it G-passive. In the following

13

all tests are performed for G = G1 that gives the largest admissible set of flightaltitudes.

6.3 Implementing the BPBAC

To implement BPBAC one needs to chose an positive definite matrix Γ and afunction φ satisfying property P(BF ). Two such functions are now exhibited.One is based on the dead-zone function ψd

f: C → C parametrized by f > 0,

such that:

ψdf(k) =

{0 (0 ≤ |k| ≤ f)(1− f

|k| )k (f ≤ |k|)

that is applied element-wize to define φdF

: Cm×l → Cm×l such that:

K = φdF(K) ⇔ K(i,j) = ψd

F (i,j)(K(i,j)) .

The second function is a smoothed version of the dead-zone ψsdf

: C → C

parametrized by f > 0, such that:

ψsdf

(k) = (|k| − f)ψdf(k)

that is applied element-wize as well to define φsdF

: Cm×l → Cm×l. For illustra-tion the two functions ψd and ψsd are plotted on Figure 3 for the real scalarcase.

Figure 3: ψd (dashed) and ψsd for f = 1

!3 !2 !1 0 1 2 3!4

!3

!2

!1

0

1

2

3

4

Lemma 1 The functions φdF

and φsdF

fulfill property P(BF ).

Proof : First, note that φ(K) = 0 for K inside the set BF is trivial. Second,recall that the scalar product is such that

Tr(φ(K)(K − F )∗) =m∑

i=1

l∑j=1

ψF (i,j)(K(i,j))(K(i,j) − F(i,j))∗

14

which is the sum over all terms of elements like ψf (k)(k− f)∗. These terms arescalar products of two vectors in the complex plane as illustrated on Figure 4.They are necessarily nonnegative. Moreover, the sum is strictly positive as soonas one element of K is outside the set BF . �

Figure 4: k − f and ψ(k) for the dead-zone and the second order dead-zone

1

R

Ik

f

f

ψ(k)

k

R

I

f

1

f2f

(k )2ψk2

ψ(k )

6.4 Simulating the BPBAC

A first series of simulations is made taking initial conditions

x(0) =(

10 0 0 0)T

, K(0) =[

0 0 0].

taking Γ = 1, F =[

10 10 10]

and φ = φdF. Robustness with respect to ξ

is illustrated for ξ = 0, ξ = 5 and ξ = 9.655. Time histories of both the outputy(t) and the control gain K(t) are plotted on Figures 5, 6 and 7 respectively.

0 1 2 3 4 5 6 7 8!40

!35

!30

!25

!20

!15

!10

!5

0

5

10

0 1 2 3 4 5 6 7 8!12

!10

!8

!6

!4

!2

0

2

4

Figure 5: y(t) and K(t) histories for ξ = 0

As expected the stabilization of the system becomes more critical as the flightaltitude is increased. Other simulations for ξ = 9.67 show that convergence tozero takes about 1000 seconds. In all cases the control gain K(t) converge to thespecified set BF as expected. This stationary value is necessarily such that theclosed-loop is G-passive. But over time, not all values of K(t) are possible SOFstabilizing gains. Indeed for ξ = 0 and ξ = 5 the time histories show temporaryunstable oscillating behaviors. These situations occur as the control gains seem

15

0 1 2 3 4 5 6 7 8−40

−35

−30

−25

−20

−15

−10

−5

0

5

10

0 1 2 3 4 5 6 7 8

−10

−5

0

5

Figure 6: y(t) and K(t) histories for ξ = 5

0 10 20 30 40 50 60 70 80 90 100−60

−40

−20

0

20

40

60

0 10 20 30 40 50 60 70 80 90 100−10

−8

−6

−4

−2

0

2

4

Figure 7: y(t) and K(t) histories for ξ = 9.655

16

to stop their convergence and their value at that time does not stabilize thelinear system. The resulting instability ”pushes” the control gain away fromthat value.

A second series of simulations is made to see the influence of the Γ matrixin particular with respect to these oscillating phenomena. The experiments aredone for ξ = 0 and for the following values of Γ:

Γ1 =

10 0 00 1 00 0 1

, Γ2 =

0.1 0 00 1 00 0 1

, Γ3 =

1 0 00 10 00 0 1

,

Γ4 =

1 0 00 0.1 00 0 1

, Γ5 =

1 0 00 1 00 0 10

, Γ6 =

1 0 00 1 00 0 0.1

.

The time histories of K(t) are represented on Figure 8. The influence of Γ isillustrated to be significant but as proved upper, any positive definite Γ makesthe system asymptotically stable.

0 5 10!15

!10

!5

0

5

10

0 5 10!15

!10

!5

0

5

10

0 5 10!15

!10

!5

0

5

10

0 5 10!15

!10

!5

0

5

10

0 5 10!15

!10

!5

0

5

10

0 5 10!15

!10

!5

0

5

10

Figure 8: K(t) histories for various values of Γ

Last series of simulations is performed for a more realistic situation withnoise on the measurements (y(t) = Cx(t)+n(t)), saturation on the inputs (u(t)is saturated between -20 and +20) and with slowly varying parameters (ξ growslinearly from 0 to 9.6 during the time interval [0 5] and then decreases linearlyto 8 at t = 10). Γ is chosen to be equal to Γ6 because it gave the best resultsin the last experiments. The time histories of y(t), K(t) and u(t) for the twochoices of dead-zone functions φd

Fand φsd

Fare plotted respectively in Figures 9

and 10.The simulations illustrate that the BPBAC has good behavior in real situa-

tions. Comparisons of these two simulations show that whatever the dead-zonefunction the closed-loop system behaves globally the same. The differences are

17

0 5 10!25

!20

!15

!10

!5

0

5

10

15

20

0 5 10!40

!35

!30

!25

!20

!15

!10

!5

0

5

0 5 10!20

!15

!10

!5

0

5

10

15

20

Figure 9: y(t), K(t) and u(t) histories in real noisy saturated situation withφ = φd

F

0 5 10!20

!15

!10

!5

0

5

10

15

0 5 10!30

!25

!20

!15

!10

!5

0

5

0 5 10!20

!15

!10

!5

0

5

10

15

20

Figure 10: y(t), K(t) and u(t) histories in real noisy saturated situation withφ = φsd

F

18

essentially in terms of convergence speed to the set BF . The second order dead-zone function φsd

Facting as a severe penalty function as the gains are far from

the set BF , it keeps the control gains closer to the set.Note that without the dead-zone functions, the controller gains would di-

verge under the influence of the noise on measurements. Indeed, as seen in thetheoretical part of the paper, the control can have infinite gain (K = −kG withany k sufficiently large) and any perturbation may tend to ”push” the gains inthat direction. This phenomenon is illustrated in Figure 11.

0 20 40 60 80 100 120 140 160 180 200!35

!30

!25

!20

!15

!10

!5

0

Figure 11: K(t) histories in real noisy saturated situation with φ = 0

7 CONCLUSIONS

Robustness problem of passification-based adaptive control is solved for the caseof linear systems with uncertainties on the A matrix. Not only robustness isproved to hold but a constructive strategy is provided to evaluate the admissibleuncertainty domains. Still several questions remain open and are left for futureresearch. The major question is whether such techniques may apply in case theB and C matrices are uncertain as well. Results in that direction are expectedprovided a relaxation of the strong HB = G∗C∗ equality constraint. Other,more practical questions are relative to the choices of the Γ matrix and the dead-zone φ function used to implement BPBAC. Several matrices Γ have been testedon the example showing very diverse convergence of the control parameters.Providing theoretical tools for choosing Γ is clearly needed. Concerning thedead-zone type functions, we have tested two such functions without noticingmuch difference. A deeper study with comparisons to other existing disturbancerejections strategies would be of interest.

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